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Criterion of existence of longitudinally inhomogeneous traveling waves in nonlinear
electrodynamics
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1978 Sov. J. Quantum Electron. 8 95
(http://iopscience.iop.org/0049-1748/8/1/L33)
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FIG. 2. Photograph of the image of
a grid with a period of 1 mm reconstructed from a hologram recorded
in a MnBi film using the wavelength
2 . 7 6 M.
MnBi. The lamps were fired 400 /i sec before a laser pulse.
The laser beam, expanded by an input collimator 9,
was split by a plane-parallel fluorite plate 10 into two
beams which were reflected by mirrors 11 and 12 and
met at an angle of 45° in the film plane.
We used this system to record the Young interference
pattern in MnBi films. An object beam incident normally on a film was reconstructed by He—Ne laser radiation (\ = 0.63 (i). The reconstructing beam was directed
at an angle a r c t , found from the relationship6
sin a,rct.
Xrcd „
sina r c d •
normally incident beam and also three interference orders of the image. This indicated that the coherence
length of the laser was at least 80 cm (i. e., the line
width was « 1/80 cm' 1 ). Thus, the CaFjtEr3* laser
emitting in the infrared at 2.76 jx could be used to record
holograms of objects with afield depth of at least 80 cm.
Thus, our investigation demonstrated that holograms
could be recorded at the wavelength of 2.76 ji, which
should be of interest in studies of various narrow-gap
semiconductors transparent in this range. The coherence length of the CaF 2 : Er3* laser was determined. The
proposed holographic method can be used to measure
the width of the emission line of this laser and also
to determine on-line the emission wavelengths of other
infrared pulse lasers.
The authors are grateful to B. M. Abakumov and
N. D. Baikova for supplying MnBi films and to Professor G. I. Rukman for his interest.
Vrct
sin
where a rcd was the angle between the reference and optic beams during the recording of a hologram (Young
interferogram); X.rcd and X.rct were the wavelengths of
the radiation used to record the hologram and to reconstruct the image, respectively. The parameters of the
system shown in Fig. 1 were Xrc<] = 2.76 [i, \ r c t = 0 . 6 3 \i,
, = 45°
The system was used also to obtain holograms of a
grid with a period of 1 mm (Fig. 2) and of a scatter
made of fluorite, which was placed in a normally incident beam.
The laser line width was estimated by recording
Young interferograms when the path difference between
the beams was between 0 and 80 cm. The reconstruction produced a point source of light corresponding to a
'M. Robinson and D.P.Devor, Appl. Phys. Lett. 10, 167 (1967).
*S. .Kh. Batygov, L. A. Kulevskii, A. M. Prokhorov, V. V.
Osiko, A. D. Savel'ev, andV. V. Smirnov, Kvantovaya
Elektron. (Moscow) 1, 2633 (1974) [Sov. J. Quantum Electron. 4, 1469 (1975)].
3
G. V. Gomelauri, L. A. Kulevskii, V. V. Osiko, A. D.
Savel'ev, andV. V. Smirnov, Kvantovaya Electron. (Moscow) 3, 628 (1976) [Sov. J. Quantum Electron. 6, 341 (1976)].
4
B. M. Abakumov, N. D. Baikova, L. N. Gnatyuk, M. L.
Gurari, and S. N. Marchenko, Kvantovaya Electron. (Moscow) 2, 1595 (1975) [Sov. J. Quantum Electron. 5, 873 (1975)].
5
B. M. Abakumov, N. D. Baikova, L. N. Gnatyuk, M. L,
Gurari, S. N. Marchenko, and G. I. Rukman, Proc. Second
Ail-Union Conf. on Photometry and Its Measuring Instruments, 1976 [in Russian].
6
R. J. Collier, C. B. Burckhardt, and L. H. Lin, Optical
Holography, Academic Press, New York (1971).
Translated by J. W. Brown
Criterion of existence of longitudinally inhomogeneous
traveling waves in nonlinear electrodynamics
A. E. Kaplan
Institute of History of Science and Technology, Academy of Sciences of the USSR, Moscow
(Submitted May 5, 1977)
Kvantovaya Elektron. (Moscow) 5, 166-168 (January 1978)
The criterion referred to in the title is established solely on the basis of the behavior of the function
e(u)u*, where u is the amplitude of the field in question and c(u) is the nonlinear permittivity of the
medium involved. The criterion is used to discuss the possibility of realization of one-dimensional
longitudinally inhomogeneous waves under various conditions and, in particular, in nonlinear optics.
PACS numbers: 42.65.Bp
In a dissipation-free medium, whose permittivity E
is a function of the field intensity IE |2 at a given point,
one-dimensional (plane) homogeneous traveling waves
of constant frequency u>0 and amplitude u can exist in
95
Sov. J. Quantum Electron. 8(1), Jan. 1978
the same way as in a linear medium:
I E M | = const = u; £(x) = uexp[i*o*VM")l, h =
0049-1748/78/0801-0095$02.40
© 1978 American Institute of Physics
(1)
95
where ko= wo/c and the amplitude-dependent phase velocity is constant. A basically new feature of the nonlinear case is the existence of longitudinally inhomogeneous traveling waves, whose amplitude and velocity
depend also on the coordinate: E(x) = u(x)exp[i / k{x)dx],
where u(x) and k(x) are real functions [the total field
is jE{x)exp(- io)0t) + c. c.]. The longitudinal aspect of
the inhomogeneity is stressed in order to distinguish
this case from the effects associated with the transverse
inhomogeneity of a wave, such as self-focusing1 and
self-bending.2 If the wave parameters vary along the
propagation axis X, the equation for the field is
iPEldx'+ fcjje (u) E = 0 (u= |£ |).
(2)
In the case of "oblique" waves (for example, in the
case when a field in a semiinfinite nonlinear medium is
excited by a wave incident from a nonlinear medium of
permittivity e0 on a plane interface between the media
and the angle of incidence is q>, measured relative to
the X axis (which is normal to the boundary and to the
polarization vector), Eq. (2) has to be modified by r e placing E(M) with £,,,= e(w) - eosinfy.
A basically new feature is the appearance of purely
traveling waves, because in other cases (when fluxes
travel in opposite directions) there are always waves
with a periodic structure of the amplitude in space,
analogous to linear standing (or partly standing) waves.
We shall define a purely traveling wave by the condition
that the wave tends to become homogeneous at infinity
where there are no radiation sources, i. e., in the
limit x - °° we have
a -*• const = uco > 0; ft (x) - • const =
= ft» > 0,
(3)
where u and k are real quantities. It should be noted
that Eq. (2) describes also the motion of a particle in
a central field and then x is the time, u is the radius,
k is the instantaneous angular velocity, and k\t is the
central potential with the opposite sign. In this case,
purely traveling waves are orbits which become circular for t — ± «=.
The waves and motion described by Eq. (2) are well
known in mechanics3 and in nonlinear electrodynamics
of plasmas [see, for example, Ref. 4, where a detailed
classification of these waves—based on the integration
of Eq. (2)—is given]. However, in the case of traveling
waves (3) we still do not know the answer to a number of
important and seemingly simple questions, the main of
which can be formulated as follows: Is it possible to use
directly the form of the function s (M) to determine the
possibility of existence of spatially inhomogeneous
traveling waves without integration of Eq. (2)? This is
an important question because integration of Eq. (2)
shows that in many typical types of nonlinearity of E(M)
when the permittivity does not pass through zero [see,
for example, Eq. (7) below], this equation admits the
existence only of homogeneous traveling waves and not
96
Sov. J. Quantum Electron. 8(1), Jan. 1978
of longitudinally inhomogeneous waves. In particular,
it is interesting that longitudinally inhomogeneous
traveling waves have not yet been found in nonlinear
optics if the same equation (2) is applied in the one-dimensional case. Therefore, the following question
arises: Are longitudinally inhomogeneous traveling
waves possible in optics at all and if the answer is in
the affirmative, under what conditions ?
We shall state and prove a comprehensive and singlevalued criterion for the existence of longitudinally inhomogeneous traveling waves, which is based only on
the behavior of the function F(M)SM4E(M) (the criterion is
formulated without proof in Ref. 5).
1. Equation (2) admits a solution of the traveling
wave type (3) if, and only if, there exists a range (U)
of values u in which c(u)>0 and if for any u in the range
(U) there is a homogeneous traveling wave solution of
the type described by Eq. (1). [This point is well-known
and trivial, and is given here for the sake of completeness; it follows from an analysis of the asymptotic behavior of k in Eq. (3)].
2. If F(u) is continuous (or piecewise continuous) and
is not falling within the range (17), then traveling wave
solutions can only be homogeneous as in Eq. (1), i. e.,
longitudinally inhomogenous traveling waves are impossible.
3. If F(u) is continuous in the range (U) and has a
continuous or a piecewise-continuous derivative, solutions in the form of longitudinally inhomogeneous traveling waves are possible if, and only if, F(u) falls at
least somewhere in (U), i . e . , if there is an interval of
monotonic fall in (17).
4. If the conditions in point 3 are satisfied and also
if the derivative dF/du is finite in (U), Eq. (2) admits
solutions in the form of longitudinally inhomogeneous
traveling waves with a given wave amplitude at infinity
MW if, and only if, M«, lies in one of the intervals (K)
where the function F{u) falls monotonically in (U).
The proof is based on an investigation of the value of
one of the integrals of motion. Writing the field in Eq.
(2) in the form E = u(x)exp[j / k(x)dx], we obtain the
first integral of motion of Eq. (2) u2k = const (energy
flux conservation) and the following equation uix):
(2a)
= 0.
Bearing in mind that in the case of traveling waves we
have k = k«u\/uz, the first integral of Eq. (2a) satisfying the traveling wave condition (3) can be obtained
in the form
'
\2
(4)
[the generalization of the laws of conservation to arbitrary field configurations and nonlinear media with e
= E(M) is made in Ref. 6].
The point 2 in the above criterion and the converse
stated in point 3 follow from the fact that if we assume
A. E. Kaplan
96
the existence of longitudinally inhomogeneous traveling
waves, i . e . , if u(x)±u* in the case of a nonfalling function F{u), we find that the right-hand side of Eq. (4)
is always nonpositive, whereas the left (with the exception of perhaps a few points) is always positive; thus,
in this case we can only have homogeneous traveling
waves (K =«„,). The direct statements in points 3 and
4 follow from the fact that if F(u) is not a nonfalling
function in (U), then ({/) should have such intervals (F)
in which dF/du < 0 and where we can always select
points M and M«, at which this derivative is finite and
continuous; then, the right-hand side of Eq. (4) is positive and this equation has a solution in the form of
longitudinally inhomogeneous traveling waves and it is
easy to show that for x — °° (and u— <*>) this solution is
of the form u(x)= «„ + Aw + o(Au), where
Au = exp [—*0(x + const)(—dF
{u«,)ldu)v'u£''].
(5)
When the conditions stated in point 4 are satisfied, such
solutions can exist for any point inside an interval (Y)
and if this interval is closed, then this applies to its end
points at which e (w J > 0 but dF(uJ/du = 0. In the latter
case if F"(uJ)±0, then inhomogeneous longitudinal
traveling waves in the limit # - °° have Aw described by
Au = —12s2,'ft§F"(u«)(x + const)2.
(6)
The converse of the conclusion in point 4 is proved in
the same way as point 2.
We shall now consider some consequences in the
above criterion.
1. If E(M) is a nonfalling function, then longitudinally
inhomogeneous traveling waves are impossible. In the
theory of nonlinear hysteretic reflection and refraction7
this means that in the case of rising e(w) we have only
the choice between homogeneous traveling waves and a
surface wave (corresponding to total reflection k=0)
in a nonlinear medium.
2. If c(u) is of the form
e (u) = e (
0),
(7)
which describes an extensive class of nonlinear mechanisms with saturation (both rising and falling right
down to zero for a = 0), then in the case of a wave incident normally on an interface between media, we
cannot have longitudinally inhomogeneous traveling
waves. They can be obtained for special types of nonnegative nonlinearities: for example, if E = E ( 0 ) / [ 1
+ p(u2i— • +M 2 ")], longitudinally homogeneous traveling
waves are possible only if w» 3.
3. In the case of normal incidence of a wave and on
condition that lAen, I «e(0) (nonlinear optics), longitudinally inhomogeneous traveling waves can be obtained
only if e (u) is in the form of a step which falls abruptly
at some value of u=u0 by an amount Ac in such a way
that the relative width of the step AM is less than its
height [AM/MO< AE/E(0)]; no mechanisms ensuring such
97
Sov. J. Quantum Electron. 8(1), Jan. 1978
nonlinearity in optics (for Au«uo) are known at present.
4. The situation changes in the case of oblique incidence of a wave near the total-internal-reflection
angle or in the case of almost glancing incidence if the
condition e(0) - e0 « c 0 is satisfied. Then, c,ff («) may
pass through zero [and, consequently, F(u) may have
a falling region] even in the case of small values of Aenl
and for ordinary types of nonlinearity. If t («) = e(0)
-ejjw2 [e 2 M 2 «e(0)]andO<e(0)-e 0 sinV s r 8 < < e(°). then
E.,f = y 2 - E2K2 and if the condition |y2«E2M20<y2 following
from point 4 of the criterion is satisfied, Eq. (2) has
solutions in the form of longitudinally inhomogeneous
traveling waves:
where B= (3wl/2-e2 -y 2 ) 1 ' 2 ; C = const (the question of the
stability of such solutions is still to be decided).
The solution ut in Eq. (8) corresponds to a wave
which initially is apparently partly "pinned" to the interface and the solution «2 behaves in the opposite manner. These waves can exist in the same range of paramaters as hysteretic jumps' in the case of falling
nonlinearity (Ref. 7 gives also estimates of the necessary parameters); their existence should clearly result
in a more complex general pattern of nonlinear reflection and refraction than in the case of a growing nonlinearity.
5. When a particle moves in a central attractive
field with a potential proportional to - 1/r truncated
fairly abruptly (for example, exponentially) at a distance ~ r 0 , we find that in the limit f — ± » we can have
trajectories with r~r0 which throughout their existence
approach the center or move away from it only once
(this is an analog of longitudinally inhomogeneous traveling waves).
The author is grateful to B. Ya. Zel'dovich, F. A.
Medvedev, and participants of a Seminar led by S. M.
Rytov for discussing the results, and to V. A. Permyakov for a valuable and stimulating discussion.
'G. A. Askar'yan, Zh. Eksp. Teor. Fiz. 42, 1567 (1962) [Sov.
Phys. JETP 15, 1088 (1962)].
A. E. Kaplan, Pis'ma Zh. Eksp. Teor. Fiz. 9, 58 (1969)
[JETP Lett. 9, 33 (1969)].
3
L. D. Landau and E. M. Lifshitz, Mechanics, 2nd ed., Pergamon Press, Oxford (1969).
4
F. G. Bass and Yu. G. Gurevich, Hot Electrons and Strong
Electromagnetic Waves in Semiconductor and Gas Discharge
Plasmas [in Russian], Nauka, Moscow (1975).
5
V. S. Butylkin, A. E. Kaplan, Yu. G. Khronopulo, andE. V.
Yakubovich, Resonant Interactions of Light with Matter [in
Russian], Nauka, Moscow (1977).
6
B. Ya. Zel'dovich, Krafk. Soobshch. Fiz. No. 5, 20 (1970).
7
A. E. Kaplan, Pis'ma Zh. Eksp. Teor. Fiz. 24, 132 (1976)
[JETP Lett. 24, 114 (1976)]; Zh. Eksp. Teor. Fiz. 72, 1710
(1977) [Sov. Phys. JETP 45, 896 (1977)].
2
Translated by A. Bryl
A. E. Kaplan
97